# The Implicitly Constructible Universe

## TL;DR

This paper explores the properties of the implicitly constructible universe, demonstrating its independence from ZFC axioms and its divergence from HOD and V, thus revealing new insights into its structure.

## Contribution

It establishes the relative consistency of Imp with the negation of CH and shows that Imp differs from HOD and V, providing new results about its foundational properties.

## Key findings

- Imp can satisfy ¬CH within ZFC consistency
- Imp is not equal to HOD in certain models
- Imp can satisfy V ≠ Imp, indicating non-absolute nature

## Abstract

We answer several questions posed by Hamkins and Leahy concerning the implicitly constructible universe, Imp. Specifically, we show that it is relatively consistent with ZFC that Imp satisfies the negation of CH, that Imp is not HOD, and that Imp satisfies V $\neq$ Imp, or in other words, that $(Imp)^{Imp}\neq Imp$.